2cos (problem 3.3.5)

Average Error: 39.6 → 0.5

Time: 6.6s

Precision: 64

Math

\[\cos \left(x + \varepsilon\right) - \cos x\]

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.033713593211140022:\\ \;\;\;\;1 \cdot \left(\log \left(e^{\cos x \cdot \left(\cos \varepsilon - 1\right)}\right) - \sin \varepsilon \cdot \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 0.03378383009651785:\\ \;\;\;\;1 \cdot \left(\cos x \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{720} \cdot {\varepsilon}^{6}\right)\right) - \sin \varepsilon \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\cos x \cdot \left(\cos \varepsilon - 1\right) - \sqrt[3]{{\left(\sin \varepsilon \cdot \sin x\right)}^{3}}\right)\\ \end{array}\]

C

double code(double x, double eps) {
	return ((double) (((double) cos(((double) (x + eps)))) - ((double) cos(x))));
}

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -0.03371359321114002)) {
		VAR = ((double) (1.0 * ((double) (((double) log(((double) exp(((double) (((double) cos(x)) * ((double) (((double) cos(eps)) - 1.0)))))))) - ((double) (((double) sin(eps)) * ((double) sin(x))))))));
	} else {
		double VAR_1;
		if ((eps <= 0.03378383009651785)) {
			VAR_1 = ((double) (1.0 * ((double) (((double) (((double) cos(x)) * ((double) (((double) (0.041666666666666664 * ((double) pow(eps, 4.0)))) - ((double) (((double) (0.5 * ((double) pow(eps, 2.0)))) + ((double) (0.001388888888888889 * ((double) pow(eps, 6.0)))))))))) - ((double) (((double) sin(eps)) * ((double) sin(x))))))));
		} else {
			VAR_1 = ((double) (1.0 * ((double) (((double) (((double) cos(x)) * ((double) (((double) cos(eps)) - 1.0)))) - ((double) cbrt(((double) pow(((double) (((double) sin(eps)) * ((double) sin(x)))), 3.0))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -0.03371359321114002

   1. Initial program 29.8

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   2. Using strategy rm

   3. Applied cos-sum 0.8

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]

   4. Applied associate--l- 0.9

      \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]

   5. Simplified 0.9

      \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\left(\sin \varepsilon \cdot \sin x + \cos x\right)}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 0.9

      \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{1 \cdot \left(\sin \varepsilon \cdot \sin x + \cos x\right)}\]

   8. Applied *-un-lft-identity 0.9

      \[\leadsto \color{blue}{1 \cdot \left(\cos x \cdot \cos \varepsilon\right)} - 1 \cdot \left(\sin \varepsilon \cdot \sin x + \cos x\right)\]

   9. Applied distribute-lft-out-- 0.9

      \[\leadsto \color{blue}{1 \cdot \left(\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)\right)}\]

   10. Simplified 0.8

       \[\leadsto 1 \cdot \color{blue}{\left(\cos x \cdot \left(\cos \varepsilon - 1\right) - \sin \varepsilon \cdot \sin x\right)}\]
   11. Using strategy rm

   12. Applied add-log-exp 0.9

       \[\leadsto 1 \cdot \left(\color{blue}{\log \left(e^{\cos x \cdot \left(\cos \varepsilon - 1\right)}\right)} - \sin \varepsilon \cdot \sin x\right)\]

   if -0.03371359321114002 < eps < 0.03378383009651785

   1. Initial program 49.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   2. Using strategy rm

   3. Applied cos-sum 48.7

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]

   4. Applied associate--l- 48.7

      \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]

   5. Simplified 48.7

      \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\left(\sin \varepsilon \cdot \sin x + \cos x\right)}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 48.7

      \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{1 \cdot \left(\sin \varepsilon \cdot \sin x + \cos x\right)}\]

   8. Applied *-un-lft-identity 48.7

      \[\leadsto \color{blue}{1 \cdot \left(\cos x \cdot \cos \varepsilon\right)} - 1 \cdot \left(\sin \varepsilon \cdot \sin x + \cos x\right)\]

   9. Applied distribute-lft-out-- 48.7

      \[\leadsto \color{blue}{1 \cdot \left(\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)\right)}\]

   10. Simplified 12.0

       \[\leadsto 1 \cdot \color{blue}{\left(\cos x \cdot \left(\cos \varepsilon - 1\right) - \sin \varepsilon \cdot \sin x\right)}\]

   11. Taylor expanded around 0 0.1

       \[\leadsto 1 \cdot \left(\cos x \cdot \color{blue}{\left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{720} \cdot {\varepsilon}^{6}\right)\right)} - \sin \varepsilon \cdot \sin x\right)\]

   if 0.03378383009651785 < eps

   1. Initial program 29.5

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   2. Using strategy rm

   3. Applied cos-sum 0.8

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]

   4. Applied associate--l- 0.8

      \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]

   5. Simplified 0.8

      \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\left(\sin \varepsilon \cdot \sin x + \cos x\right)}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 0.8

      \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{1 \cdot \left(\sin \varepsilon \cdot \sin x + \cos x\right)}\]

   8. Applied *-un-lft-identity 0.8

      \[\leadsto \color{blue}{1 \cdot \left(\cos x \cdot \cos \varepsilon\right)} - 1 \cdot \left(\sin \varepsilon \cdot \sin x + \cos x\right)\]

   9. Applied distribute-lft-out-- 0.8

      \[\leadsto \color{blue}{1 \cdot \left(\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)\right)}\]

   10. Simplified 0.8

       \[\leadsto 1 \cdot \color{blue}{\left(\cos x \cdot \left(\cos \varepsilon - 1\right) - \sin \varepsilon \cdot \sin x\right)}\]
   11. Using strategy rm

   12. Applied add-cbrt-cube 0.9

       \[\leadsto 1 \cdot \left(\cos x \cdot \left(\cos \varepsilon - 1\right) - \sin \varepsilon \cdot \color{blue}{\sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}}\right)\]

   13. Applied add-cbrt-cube 0.9

       \[\leadsto 1 \cdot \left(\cos x \cdot \left(\cos \varepsilon - 1\right) - \color{blue}{\sqrt[3]{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}} \cdot \sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}\right)\]

   14. Applied cbrt-unprod 0.9

       \[\leadsto 1 \cdot \left(\cos x \cdot \left(\cos \varepsilon - 1\right) - \color{blue}{\sqrt[3]{\left(\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon\right) \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right)}}\right)\]

   15. Simplified 0.9

       \[\leadsto 1 \cdot \left(\cos x \cdot \left(\cos \varepsilon - 1\right) - \sqrt[3]{\color{blue}{{\left(\sin \varepsilon \cdot \sin x\right)}^{3}}}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.033713593211140022:\\ \;\;\;\;1 \cdot \left(\log \left(e^{\cos x \cdot \left(\cos \varepsilon - 1\right)}\right) - \sin \varepsilon \cdot \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 0.03378383009651785:\\ \;\;\;\;1 \cdot \left(\cos x \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{720} \cdot {\varepsilon}^{6}\right)\right) - \sin \varepsilon \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\cos x \cdot \left(\cos \varepsilon - 1\right) - \sqrt[3]{{\left(\sin \varepsilon \cdot \sin x\right)}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))